Algebra (from Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in
x
+
2
=
5
{\displaystyle x+2=5}
the letter
x
{\displaystyle x}
is an unknown, but applying additive inverses can reveal its value:
x
=
3
{\displaystyle x=3}
. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
A mathematician who does research in algebra is called an algebraist.
Hello! I am sorry that this questions is not actually directly related to physics, but, can anyone recommend me a good book about abstract algebra (basically lie algebra, representation theory etc.) used in physics? I have tried for a long time to find something online but I haven't find a...
Homework Statement
Note: I'm saying it's very very hard because I still couldn't solve it and I've posted it in stackexchange and no answer till now.
I'm posting here the problem statement, all variables and known data in addition to my solving attempts. Because I'm posting an image of my...
hi! i need help for this inequality
1. ##a\in\mathbb{N}*~and~ \frac{a}{a+1}<\frac{a+1}{a+2}<\frac{a+2}{a+3}##
show that : ##\frac{1}{2}*\frac{4}{5}*...*\frac{2005}{2006}*\frac{2008}{2009}<\frac{1}{12}##
Here i have stoped. Please tell me if is corect what i have done so far and how to continue ...
Homework Statement
Consider the four vectors (1, 1, 1), (2, −1, 3), (1, 7, −1) and (1, 4, 0). Calculate how many ways you can write (1, 1, 1) as a linear combination of the other three, explaining your reasoning.
The Attempt at a Solution
wouldn't any of these combinations give the correct...
Homework Statement
(A OR C) AND NOT(C AND A AND B OR C AND A AND NOT B)
or
(A + C) (CAB + CAB')'
Relevant Equations
(A+B)' = A'B'
A(B+C) = (AB) + (AC)
(AB)' = A' + B'
The attempt at a solution
I'm not sure how I'm suppose to expand (CAB + CAB')' for simplifying. I keep arriving at false which...
The problem statement: Show that if ##r_1## and ##r_2## are the distinct real roots of ##x^2 + px + 8 = 0##, then ##r_1 + r_2 > 4 \sqrt{2}##.
We start by noting that ##r_1 r_2 = 8##. Using this relation, we'll find the minimum value of ##r_1 + r_2##. To minimize ##r_1 + r_2##, we need to...
Hello, I just completed a first course in linear algebra and really enjoyed my studies. So much so that I want to pursue it more in the fall as an independent study, i am a EE major in college and was curious what directions might be useful for applications in that field.
Dear Physics Forums friends,
I am an aspiring mathematician who is deeply interested in the analysis, topology, and their applications to the microbiology. Recently, I started to become very curious about why concepts and theorems in the real analysis and topics come as they are; the...
Question 1:
Consider the numbers 2 & 8. The average between these two quantities is 5, hence 2+8=10, 10/2=5. Now consider two arbitrary derivatives. It wouldn't make much sense to find the average between two unrelated derivatives, but suppose that f(x,y) was a function of both x & y. Now...
Homework Statement
I did some refresher exercises in khan academy. I tried to wrap my head around the factoring problems, but it did not go well...
I did not understand the khan academy tips and hints. (about factoring)
one particular difficult polynomial was thus: factorize the polynomial...
Homework Statement
Homework Equations
a) the one given
b) det(A-λI) = 0
find λ values using A
c)use λ values to find eigenvectors
The Attempt at a Solution
This wasn't explained well enough so I can understand it in class. So far, I made the matrix being multiplied to A have the following...
Homework Statement
Prove Rθ+φ =Rθ+Rφ Where Rθ is equal to the 2x2 rotational matrix
[cos(θ) sin(θ),
-sin(θ) cos(θ)]
Homework Equations
I am having a hard time trying figure our what is being asked. My question is can anyone put this into words? I am having trouble understanding what the phi...
Homework Statement
Let P2 be the vector space of all polynomials of a degree at most 2 with real coefficients. Let T: P2→ℝ be the functioned defined by:
##T(p(t)) = p(2) - p(1)##
a) Find a non-zero element of the Kernel of T. (I think I figured this one out, but I'm not too sure).
b) Find a...
Hey everyone, I understand how to normalize a second order system, but I wanted to know if the same steps are taken when the parameters of the system are not scalar but matrices. For example
where the parameter phi, and gamma are both 3x3 matrices and X is a 3x1 vector.
From what I've see...
Hello from Italy
I'm switching from CS to Physics BS because i personally find it more various and interesting (and in Rome there is one of the best physics school in the world).
Mathematical analysis is a common subject and my credits will be recognised but i didn't study Linear algebra yet...
Homework Statement
Homework Equations
Quadratic discriminant
The Attempt at a Solution
I did b^2 -4ac>=0, but the algebra becomes prohibitively difficult.
I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby.
They claim in chapter 4 that the geometric product ab between two vectors a and b is defined according to the axioms
i) associativity: (ab)c = a(bc) = abc
ii) distributive over addition: a(b+c) = ab+ac
iii) The...
What are good books in universal algebra, given that I have a background in Herstein (Topics in Algebra), Hubbard/Hubbard, Engelking (Topology), and Dugundji (Topology)? I am currently reading Hungerford, and I found a field called universal algebra while searching internet for some concepts...
Hi all,
Just curious on whether temporal databases are designed in the same way as "standard" relational databases,
so that queries on temporal databases are wffs in temporal logic , i.e., using Until, Release and all other unary
operators , and the query outputs is the set of models for the...
Hi
As the title suggests, I'm looking for a math book that has some competition-style problems in it, with explanations on how to solve them. I have found numerous of other competitions online but the explanations are very not understandable for me. I am NOT talking about international...
Homework Statement
1. How did they complete the square for these equations in the picture below? What was their thought process?
2. distance/velocity = time , velocity/acceleration = time , In leibniz notation how does this cancel out?
When you divide, how does it cancel out to give you a...
micromass submitted a new PF Insights post
How to self-study algebra. Part II: Abstract Algebra
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Continue reading the Original PF Insights Post.
Homework Statement
If n is a positive integer, then 2x2 matrix [-32,252] [-4,32] raised to the power of n is...
Homework Equations
I know that first I should diagonalize the given matrix, something I also seem to have a hard time with.
The Attempt at a Solution
I determined the eigenvalues...
So I've always been fairly good at whatever science I learned in elementary and middleschool (which was mostly biology), and I've fairly enjoyed it, but I've only recently started really learning, and I've come to love it. I've especially found quantum physics interesting, however, seeing as I...
Homework Statement
Find the matrix that represents a rotation counterclockwise around the origin by 75 degrees followed by a reflection about the x-axis
Homework Equations
I know that for A rotated counter clockwise you use the 2x2 matirx [cos(theta), -sin(theta)] [sin(theta, cos(theta)] and...
Hi everyone, didn't know where to post question on sigma algebra so here it is:-
What I've tried till now:
Let C\in G
1) For C=X, f^{-1}(B)=X which will be true for B=Y (by definition)
2) For closure under complementation, to show C^{c}\in G. So, C^{c}=X\setminus C=X\setminus...
Current schedule:
Summer: Elective
Fall: -not important, but it can't be changed in any way-
Spring: Differential Eqs., Classical Mechanics & Mathematical Methods 1 (Physics), Linear Algebra, Chemistry Lab, Elective
I'm thinking that since the spring schedule looks kinda heavy I can switch up...
Homework Statement
The period of a simple pendulum, defined as the time necessary for one complete oscillation, is measured in time units and is given by
T = 2π ℓ/g
where ℓ is the length of the pendulum and g is the acceleration due to gravity, in units of length divided by time squared. Show...
I am reading through a introductory algebra textbook and refreshing memory on this topic. In the matter of the Associative Properties, it is giving me the expression 3 + (8+x).
(I'm skipping the graphs section because I'm leaving that for my tutor Saturday. But I'm moving to algebra because I...
I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ...
On page 657 D&F give a definition of a k-algebra ... as follows:
I have to say I do not find that this definition gives me a good intuitive idea...
I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ...
On page 657 D&F give a definition of a k-algebra ... as follows:I have to say I do not find that this definition gives me a good intuitive idea...
Hello! Please help continue solving tye problem I got stuck.
This is my attemptlet $d=$ length of the circuit
$\frac{60}{a}$mph --- speed for walking
$\frac{60}{b}$mph ----speed for riding
$\frac{60}{c}$mph ---- speed for driving
$d = \frac{60}{a}t_{1}+ \frac{60}{b}t_{2} + \frac{60}{c}t_{3}$...
Homework Statement
Homework Equations
I can't start because I did not understand the statement in red underline. Please explain to me first. Then I may be able to solve it. Sorry english is not my first language. Hope you would understand. Thank you!
The Attempt at a Solution
Hello!
I am a diploma student at HEP section. I am going to have an interview for PhD within a week. I've finished the course and learned a lot about Lie algebra, quantum field theory, general relativity, standard model, etc. How can I review everything as soon as possible? For example, Mark...
https://scontent-mxp1-1.xx.fbcdn.net/v/t35.0-12/13324008_10209600482268675_3779883_o.png?oh=abd4c815b9472f70cfbfbb1923a37b3c&oe=5752F454
Number 12 : I have simplified it and I have got AB but the model answer in red say A(comp(B)+c)
I just want is that answer true or not , thanks in advance
MENTOR NOTE: Moved here from the General Math forum hence no homework template.
Hello! Sorry for not showing what I have tried to solve the problem.
Here it is
I let x = amount of inferior tea to be added to superior tea in lbs.
y = amount of superior tea in lbs.
3x+5y = 11/3(x+y)...
##\nabla p = \rho \nabla \phi ##
My textbook says that by taking the curl we get:
## 0=\nabla \rho X \nabla \phi ## **
I don't follow. I understand the LHS is zero, by taking the curl of a divergence.
But I'm unsure as to how we get it into this form, from which it is clear that the gradients...
Homework Statement
Under what restrictions on ##c, d, e##, will the combinations ##c\vec u + d\vec v + e\vec w## fill in the dashed
triangle?
Homework Equations
The Attempt at a Solution
Clearly, ##\vec w + a (\vec v - \vec w) + b(\vec u - \vec v)## will be in the triangle when ##0 \leq b...
I'm trying to better understand RSA and Diffie-Hellman exchange and the modulus math that they are base on, there are some questions I have about there properties for which I am unable to find concise explanations about. I'm generally interested in how the commutative, associative...
Homework Statement
son and father and granpa have same value for their birthday (except the year ostensibly)
father is nine times the son's age now. calculate the granpa's age now.
clues
- five years ago, grandpa was 32 years older than what the father was at five years ago.
-in five years...
Homework Statement
cellphones are sold at price of 302$ per unit
In one week there were 450 units sold.
for every one dollar increase in unit price, the weekly sales of the phones fell by four units.
form a function which describes the weekly sales income when the phone's price changes by X...